Analytical methods are one of the most important approaches to the image reconstruction from projections problem (see e.g. [1] G. N. Ramachandran, A. V. Lakshminarayanan, Three-dimensional reconstruction from radiographs and electron micrographs: II. Application of convolutions instead of Fourier transforms, Proc. Nat. Acad. Sci. of USA, vol. 68, pp. 2236-2240, 1971, [2] R. M. Lewitt, Reconstruction algorithms: transform methods, Proceeding of the IEEE, vol. 71, no. 3, pp. 390-408, 1983). Another major category of reconstruction method is the algebraic reconstruction technique (ART) (see e.g. [3] S. Kaczmarz, Angeneaherte Aufloesung von Systemen Linearer Gleichungen, Bull. Acad. Polon. Sci. Lett. A., vol. 35, pp. 355-357, 1937, [4] Y. Censor, Finite series-expansion reconstruction methods, Proceeding of the IEEE, vol. 71, no. 3, pp. 409-419, 1983). All of the recent practically applicable reconstruction algorithms can be classified as belonging to one of these two methodologies of image reconstruction.
In conventional tomography, analytical algorithms, especially those based on convolution and back-projection strategies of image processing, are the most popular. Algebraic algorithms are much less popular because algebraic reconstruction problems are formulated using matrices with very large dimensionality. Thus algebraic reconstruction algorithms are much more complex than analytical methods.
Recently, there have been some new concepts regarding reconstruction algorithms. Among these new ideas, the statistical approach to image reconstruction is preferred (see e.g. [5] K. Sauer, C. Bouman, A local update strategy for iterative reconstruction from projections, IEEE Transactions on Signal Processing, vol. 41, No. 3, pp. 534-548, 1993, [6] C. A. Bouman, K. Sauer, A unified approach to statistical tomography using coordinate descent optimization, IEEE Transactions on Image Processing, vol. 5, No. 3, pp. 480-492, 1996). This concept has been adapted for three-dimensional multi-slice helical computed tomography (see e.g. [7] J.-B Thibault, K. D. Sauer, C. A. Bouman, J. Hsieh, A three-dimensional statistical approach to improved image quality for multi-slice helical CT, Medical Physics, vol. 34, No. 11, pp. 4526-4544, 2007) as the iterative coordinate descent (ICD) approach. In the ICD algorithm, the reconstruction process is performed using the maximum a posteriori probability (MAP) approach formulated principal as an algebraic reconstruction problem. This methodology is presented in the literature as being more robust and flexible than analytical inversion methods because it allows for accurate modeling of the statistics of projection data.
The present applicant furnishes a new statistical approach to the image reconstruction problem, which is consistent with the analytical methodology of image processing during the reconstruction process. The preliminary conception of this kind of image reconstruction from projections strategy is represented in the literature only in the original works published by the present applicant for parallel scanner geometry (see e.g. [8] R. Cierniak, A novel approach to image reconstruction from projections using Hopfield-type neural network, Lecture Notes in Artificial Intelligence 4029, pp. 890-898, Springer Verlag, 2006, [9] R. Cierniak, A new approach to image reconstruction from projections problem using a recurrent neural network, International Journal of Applied Mathematics and Computer Science, vol. 183, No. 2, pp. 147-157, 2008, or [10] R. Cierniak, A new approach to tomographic image reconstruction using a Hopfield-type neural network, International Journal Artificial Intelligence in Medicine, vol. 43, No. 2, pp. 113-125, 2008), for fan-beam geometry (see e.g. [11] R. Cierniak, New neural network algorithm for image reconstruction from fan-beam projections, Neurocomputing, vol. 72, pp. 3238-3244, 2009) and for spiral cone-beam tomography (see e.g. [12] R. Cierniak, A three-dimensional neural network based approach to the image reconstruction from projections problem, Lecture Notes in Artificial Intelligence 6113, S. 505-514, Springer Verlag, 2010). In all these algorithms the reconstruction problem for any geometry of scanner is reformulated to the parallel-beam reconstruction problem using a re-binning operation. Thanks to the analytical origins of the reconstruction method proposed in the above documents, most of the above-mentioned difficulties connected with using ART methodology can be avoided.